It is usually the case that studios, agents and distributors will co-ordinate release dates to capitalise on a star's popularity, but it is rare that an A-list actress will be headlining two remarkable new films and an international political controversy all at the same time. That's not your run-of-the-mill Hollywood career, but then Scarlett Johansson is no common-or-garden star.

The nature of her latest movies tells you as much. In Her, which has just opened in the UK, she plays Samantha, a computer operating system that evolves in response to the user's needs and personality. She is as present as any of her flesh-and-blood co-stars despite appearing in voice only. Her is complimented by Under the Skin, in which she is seen but rarely heard. As an extra-terrestrial who assumes human form to cruise the streets of Glasgow in a Transit van looking for men to devour, she has perhaps 20 or 30 lines in the entire film. She is almost unrecognisable in a brown wig and a ratty fur coat, though on closer inspection the plump, amused lips are a giveaway.

Under the Skin, released next month, was made under deeply unorthodox conditions that attest to Johansson's flexibility and daring. For the scenes in which she is driving around Glasgow, the director, Jonathan Glazer, rigged her vehicle with tiny cameras so that she could be filmed surreptitiously picking up her "prey"– real people, rather than actors – who Johansson selected spontaneously as she was driving.

The unsuspecting men would be informed by a crew member that they had just been filmed and then asked to sign a release form. Those who needed extra convincing might be told the identity of the mystery driver who had just chatted to them or driven them a few miles down the road. "Good gosh!" would come the reply, or a close equivalent in the Glaswegian vernacular.

It's hard to think of many stars of Johansson's stature who would throw themselves willingly into such an inherently suspenseful exercise, incorporating that level of risk and that demand for improvisation. In a business where anyone vaguely pleasing to the eye can rise to the top by hitting their marks and smiling on the red carpet, Johansson goes several extra miles, in this instance, while driving on the left in a Transit van.

Some people at Oxfam, where she has been a goodwill ambassador from 2005 until last month, might be wishing that her near-silence had extended from the set of Under the Skin and into her public life. When Johansson signed up to be the face of SodaStream, beginning with a half-time commercial at the recent Super Bowl, Oxfam expressed concern over the implicit conflict: the carbonated drinks company has a factory in an Israeli settlement in the West Bank.

"Oxfam is a human rights organisation," observed the boycott campaign founder, Omar Barghouti. "They cannot maintain an ambassador if they are involved in a complicit Israeli company built in a settlement." Sure enough, Johansson stepped down from Oxfam three weeks after the row began, amid accusations that she had been politically naive. But it was the tenor of her statement that alarmed some commentators.

"Scarlett Johansson has respectfully decided to end her ambassador role with Oxfam after eight years," her spokesperson said. "She and Oxfam have a fundamental difference of opinion in regards to the boycott, divestment and sanctions movement. She is very proud of her accomplishments and fundraising efforts during her tenure with Oxfam."

A commentator on the opinions website Mondoweiss described this as Johansson not only extricating herself from a tricky position but "cynically and opportunistically… throwing Oxfam under the bus" by suggesting that she had been asked implicitly to boycott SodaStream, when, in fact, the charity only wanted her to refrain from promoting the firm. The Mondoweiss writer worried that this leaves Oxfam vulnerable to accusations that it supports a boycott of Israel, when, in fact, it does not express any opinion.

If there is an upside to this argument, it must be that it has dragged a contentious political issue into the pop-culture spotlight; refracted through Johansson's celebrity, it suddenly enters the lives and perspectives of her fans. How it reflects on Johansson is another matter. She has not been slow to exhibit exasperation with the PR treadmill, but getting quizzed at press junkets about, say, your character's motivation in Iron Man 2 must have looked preferable over the last month to having aspersions cast on your political integrity.

As a child, Johansson's mother took her and her twin brother, Hunter, with her whenever she voted in order to impress upon them the importance of the democratic process. "Both of my parents were extremely liberal," she has said, "so I think I grew up in a household that was very politically conscious. We all watched the elections on TV and we watched the debates. We just naturally became politically active. It was just understood that it was important, that it was our responsibility. I never tell people whom to vote for. I'm not telling people where to give money, but if there is to be a spotlight shed on me, then I'd like to direct that spotlight on to causes I think are worthy or on to interesting, progressive figures." Or, for that matter, manufacturers of fizzy drinks.

Johansson has had an unimpeachable career so far. That is not to say she has played it safe. Though she has been in her fair share of duds (The Nanny Diaries, Michael Bay's The Island), it would be a hard-hearted soul who placed the blame for them at her feet. Her three-film collaboration with Woody Allen (beginning with Match Point, in which she was cast after the original choice, Kate Winslet, dropped out) included the regrettable Scoop, which didn't get a cinema release in the UK, but then it also produced the divine Vicky Cristina Barcelona. Swings and roundabouts.

What makes her such an exhilarating performer is her apparent derision for the very celebrity culture in which she is embroiled. That impression is partly an accident of physiognomy and geography – that slight Elvis sneer in her smile, which signals a pre-emptive sarcasm, coupled with the vocal twang inherited from her Bronx-born mother. But from her earliest performances, there was a sense that Johansson was on to something; she was a step ahead of her peers. As far back as Ghost World, an adaptation of Daniel Clowes's graphic novel in which she played, at the age of 17, one half of a pair of sardonic outsiders, she exuded an inner knowingness.

That has survived into her work today, whether it is her luminous performance in Girl With a Pearl Earring, strange, small-scale productions such as the porn-addict comedy Don Jon or superhero extravaganzas such as Avengers Assemble, in which she played Black Widow, essentially Rita Hayworth with a knack for martial arts. Like Jennifer Lawrence, she combines the lustre of a star with the earthy, unpredictable frisson of a character actor.

It was only two years after Ghost World that she got one of her key roles: the lead in Sofia Coppola's Lost in Translation, a plaintive, semi-romance about two American misfits at a plush Tokyo hotel. Johansson was the young woman left to her own melancholia while her photographer boyfriend darted off on assignments, while Bill Murray was the ageing actor with whom she makes an unexpected connection.

The film had its share of easy reassurances, but Johansson was a revelation. Much of the movie hinged on close-ups of her pensive, enigmatic beauty, her ability to convey deep reserves of intelligence and sadness.

Though woe betide anyone who took her in real life to be the ethereal creature she is in the movie. Journalists who presumed to ask what Bill Murray's character whispers in her ear at the end of the film could bank on an acidic dismissal: "Oh, you think you're the first one to ask?" she replied to one reporter.

Until the fall-out over SodaStream, her only roundly negative experiences with the media had been the result of prurient journalism – it wasn't her fault that the US press was fascinated by her divorce after three years of marriage to the actor Ryan Reynolds, or that her computer was hacked and nude photographs of her distributed online.

If the SodaStream affair has taken a little of the fizz out of her persona, that shouldn't diminish the artistic choices she has made. Hollywood stars are always pimping themselves out to brands and companies: that was one of the running gags of Lost in Translation. What makes Johansson different from most celebrities who shill for the highest bidder is that it doesn't quite square with the coolness she has cultivated so far. She should be pleased about that. She might even take the furore as a bitter kind of compliment.